Interaction between heterogeneously charged surfaces: Surface patches and charge modulation

Dan Ben-Yaakov, David Andelman, and Haim Diamant

Phys. Rev. E 87, 022402 – Published 5 February 2013

Abstract

When solid surfaces are immersed in aqueous solutions, some of their charges can dissociate and leave behind charged patches on the surface. Although the charges are distributed heterogeneously on the surface, most of the theoretical models treat them as homogeneous. For overall non-neutral surfaces, the assumption of surface charge homogeneity is rather reasonable since the leading terms of two such interacting surfaces depend on the nonzero average charge. However, for overall neutral surfaces the nature of the surface charge distribution is crucial in determining the intersurface interaction. In the present work we study the interaction between two charged surfaces across an aqueous solution for several charge distributions. The analysis is preformed within the framework of the linearized Poisson-Boltzmann theory. For periodic charge distributions the interaction is found to be repulsive at small separations, unless the two surface distributions are completely out-of-phase with respect to each other. For quenched random charge distributions we find that due to the presence of the ionic solution in between the surfaces, the intersurface repulsion dominates over the attraction in the linear regime of the Poisson-Boltzmann theory. The effect of quenched charge heterogeneity is found to be particularly substantial in the case of large charged domains.

Received 12 May 2012

DOI:https://doi.org/10.1103/PhysRevE.87.022402

Authors & Affiliations

Dan Ben-Yaakov and David Andelman^*

Raymond & Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

Haim Diamant

Raymond & Beverly Sackler School of Chemistry, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

^*andelman@post.tau.ac.il

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Vol. 87, Iss. 2 — February 2013

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Figure 1
Schematic drawing of two planar heterogeneously charged surfaces immersed in ionic solution. The gray regions are positively charged, and the white ones are negatively charged. The charge distributions of the bottom and top surfaces are given by $Σ_{B} (x, y)$ and $Σ_{T} (x, y)$ , respectively. The intersurface separation is $d$ .Reuse & Permissions
Figure 2
The intersurface interaction (per unit area) $\hat{F} / A$ for a single $q_{0}$ mode, rescaled by the prefactor of Eq. (15), $\hat{F} = (q_{0} / κ_{_{D}} C_{+}^{2}) F$ , and plotted for the equal amplitude case: $C_{σ} = C_{η} = C_{+}$ ( $C_{-} = 0$ ). (a) The intersurface interaction as a function of the intersurface separation $d$ in units of $q_{0}^{- 1}$ . The values of the relative angle $φ$ are 0, $0.85 π$ , $π$ for the solid, dashed, and dotted lines, respectively. (b) The intersurface interaction as a function of the relative phase $φ$ . The values of $q_{0} d$ are 0.5, 1, and 2 for the solid, dashed, and dotted lines, respectively.Reuse & Permissions
Figure 3
The averaged interaction energy, $\hat{F} / A$ as a function of the intersurface separation $d$ for randomly charged surfaces. (a) The free energy from Eq. (33), for an uncorrelated Gaussian distribution [Eq. (32)], is plotted on a semilog plot as a function of $κ_{_{D}} d$ . (b) The free energy is plotted on a semilog plot as a function of $κ_{_{D}} d$ , where the charge distribution is a two-dimensional Lorentzian, Eq. (36). Three values of $κ_{_{D}} ξ$ are shown: $κ_{_{D}} ξ = 100$ (solid line), $κ_{_{D}} ξ = 1$ (dashed line), and $κ_{_{D}} ξ = 0.1$ (dotted line). In the inset the dependence of the free energy on $κ_{_{D}} ξ$ is plotted on a log-log scale for two values of the separation: $κ_{_{D}} d = 0.1$ (solid line) and $κ_{_{D}} d = 1$ (dashed line). The plotted free energy is rescaled by the prefactor of Eq. (33), $\hat{F} = 2 π 〈 F 〉 / (κ_{_{D}}^{2} γ^{2} a^{2})$ for (a), and $\hat{F} = 〈 F 〉 / γ^{2}$ from Eq. (36) for (b).Reuse & Permissions
Figure 4
The patch-patch intersurface interaction energy per unit area, $| \hat{F} | / A$ , as a function of the separation $d$ rescaled by $κ_{_{D}}$ . The plotted free energy $\hat{F} = F / κ_{_{D}}^{2}$ is rescaled by $κ_{_{D}}^{2}$ . (a) Attraction $| {\hat{F}}_{att} |$ (dashed line) and repulsion ${\hat{F}}_{rep}$ (solid line) are calculated by the nonlinear PB equation for $σ = - η = 10 κ_{_{D}}$ and $σ = η = 10 κ_{_{D}}$ , respectively. The inset shows the average of the repulsive and attractive corresponding free energies ${\hat{F}}_{av} = ({\hat{F}}_{rep} - | {\hat{F}}_{att} |) / 2$ , calculated by the linear and nonlinear PB equation. (b) The solid and dashed lines are calculated with the nonlinear PB equation for $σ = η = 0.5 κ_{_{D}}$ and $σ = - η = 0.5 κ_{_{D}}$ , respectively. The inset shows the average ${\hat{F}}_{av}$ . The results of the linear PB equation are omitted in (b) since their difference from the nonlinear calculation is invisible.Reuse & Permissions

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